有向图的外独立双罗马控制

Outer independent double Roman domination of a digraph

设D=(V(D),A(D))是一个有向图,若函数h:V(D)→{0,1,2,3}满足以下条件:(1)每个赋值为0的顶点至少有一个赋值为3的内邻或两个赋值为2的内邻;(2)每个赋值为1的顶点至少有一个赋值为2或3的内邻;(3)所有赋值为0的顶点都是不相邻的,则称函数h为有向图D的一个外独立双罗马控制函数.一个有向图的外独立双罗马控制函数的权为所有顶点的赋值之和.外独立双罗马控制函数的最小权称为外独立双罗马控制数.2020年,Abdollahzadeh Ahangar等人给出了无向图的外独立双罗马控制函数的定义,并给出了无向图外独立双罗马控制数的一些界.文中将此概念推广到了有向图上,研究了有向图的外独立双罗马控制数的界,并进一步刻画了外树的外独立双罗马控制数的下界,以及外独立双罗马控制数的Nordhaus-Gaddum不等式.

Let D be a digraph.An outer independent double Roman dominating function on D is a function h:V(D)→(0,1,2,3)satistfying the following conditions:(1)each vertex with label 0 is adjacent to at least one in-neighbour with label 3 or two in-neighbours with label 2;(2)each vertex with label 1is adjacent to at least one in-neighbor with label greater than 1;(3)all vertices labeled by O is independent.The weight of an outer independent double Roman dominating function is the sum of assigned values of all vertices.The outer independent double Roman domination number is defined as the minimum weight of an outer independent double Roman dominating function on D.In 2020,the concept and various bounds of the outer independent double Roman dominating function on agraph was provided by Abdollahzadeh Ahangar et al.In this paper,this concept is extended to digraph and the bounds of the outer independent double Roman dominating function of a digraph are studied.Furthermore,the lower bound of the outer independent double Roman domination number of an out-tree and the Nordhaus-Gaddum inequality for the outer independent double Roman domination number of a digraph are characterized.